Additive commutators and trace over a PID I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{M}_n(R)$, $A\neq BC-CB$).
I know that such a PID (if it can be found) cannot be a field, or $\mathbb{Z}$.
 A: Every matrix with trace zero over a PID is a commutator, according to the MR review of
Rosset, Myriam(IL-BILN); Rosset, Shmuel(IL-TLAV)
Elements of trace zero that are not commutators. 
Comm. Algebra 28 (2000), no. 6, 3059--3072. 
From the Math Review:
Although Shoda's method fails when $C$ is a PID, the authors do prove the result in this case, and give counterexamples for $C$ of dimension $\ge 2$.
However, I just took a look at the paper, and as far as I can see the authors only claim the result for 2x2 matrices!
Can anyone resolve this conundrum?
A: Here (see the very last paragraph) it is stated that every matrix with trace zero over a PID is a commutator. However, I can't come up with a proof right away; the only proof for matrices over a field that I remember (due to Albert?) does not immediately generalize. 
A: I'm a bit (?) late on that one, but the theorem for general PIDs has been proven after this question was asked.
The reference is (see here for the ArXiv version):

Stasinski, Alexander, Similarity and commutators of matrices over principal ideal rings, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2333–2354.

A: It is not difficult to see that Rosset & Rosset's result for $2\times2$ matrices is equivalent to the surjectivity of the bilinear map $(X,Y)\mapsto X\times Y$ (called vector product when $A={\mathbb R}^3$) over $A^3$. For this, just search $B$ and $C$ such that $b_{22}=c_{22}=0$.
To prove it, let $Z=(a,b,c)\in A^3$ be given. One can choose a primitive vector $X=(x,y,z)$ such that $ax+by+cz=0$. By primitive, I mean that $gcd(x,y,z)=1$. Bézout tells that there exist a vector $U=(u,v,w)$ such that $ux+vy+wz=1$. Set $Y=Z\times U$. Then $Z=X\times Y$. 
